Independent-cluster parametrizations of wave functions in model field theories. I. Introduction to their holomorphic representations

Abstract

The configuration-interaction method (CIM), normal coupled-cluster method (NCCM), and extended coupled-cluster method (ECCM) form a rather natural hierarchy of formulations of increasing sophistication for describing interacting systems of quantum-mechanical particles or fields. They are denoted generically as independent-cluster (IC) parametrizations in view of the way in which they incorporate the many-body correlations via sets of amplitudes that describe the various correlated clusters within the interacting system as mutually independent entities. They differ primarily by the way in which they incorporate the exact locality and separability properties. Each method is shown to provide, in principle, an exact mapping of the original quantum-mechanical problem into a corresponding classical Hamiltonian mechanics in terms of a set of multiconfigurational canonical field amplitudes. In perturbation-theoretic terms the IC methods incorporate infinite classes of diagrams at each order of approximation. The diagrams differ in their connectivity or linkedness properties. The structure of the ECCM in particular makes it capable of describing such phenomena as phase transitions, spontaneous symmetry breaking, and topological states. We address such fundamentally important questions as the existence and convergence properties of the three IC parametrizations by formulating the holomorphic representation of each one for the class of single-mode bosonic field theories which include the anharmonic oscillators. These highly nontrivial models provide a stringent test for the coupled-cluster methods. We present a particularly detailed analysis of the asymptotic behaviour of the various amplitudes which exactly characterize each IC method. More generally, the holomorphic representation allows us to give a completely algebraic description of all aspects of each scheme. In particular, this includes the topological connectivity properties of the various terms or diagrams in their expansions. We construct a generating functional for the calculation of the expectation values of arbitrary operators for each of the IC parametrizations. The functional is used in each case to formulate the quantum mechanical action principle and to perform the mapping into the corresponding classical phase space.